I was trying to figure out all the various methods of the basic skill building (unmodded, no cheats, and no reward track items) and at what age they can be learned. This is what I came up with. Did I get everything? Is the mirror really the only way to learn charisma?

Also, what will they do autonomously (again unmodded or cheats)? All listed after (auto) are ones I know/ have seen sims do as such.

Body is learned by - exercising (tv or machine), yoga, jogging, (auto) swimming, electro-dance-sphere, football, basketball, bowling, jumping rope, or playground toys, .
Can be learned by child - elder.

Charisma is learned by - practicing (romance/ talking?) in a mirror, sing nursery rhyme together, or talking to the parrot; (auto)spin bunny - toddler only.
Can be learned by toddler - elder.

Cleaning is learned by - studying, raking leaves, filling in dirt holes, or (auto) cleaning objects.
Can be learned by child - elder.

Cooking is learned by - studying, watching cooking channel, or cooking.
Can be learned by child - elder.

Creativity is learned by - writing a novel, (auto) playing an instrument, or painting; (auto) xylophone - toddler only.
Can be learned by toddler - elder.

Logic is learned by - newspaper crossword, (auto) playing chess or, using the telescope; (auto) shape sorter - toddler only.
Can be learned by toddler - elder.

Mechanical is learned by - studying, tinkering with objects, or fixing broken objects; (auto) activity table - toddler and children only.
Can be learned by toddler - elder.

The newspaper crossword builds logic too (albeit really really slowly, since you just get one a day!)

There are all the different career and hobby chance cards that have an impact on skills too...

Yoga also for body

Charisma can be gained by talking to the parrot, and children can do it.

Children can use the activity table, and it builds both mechanical and creativity.

Yoga, football, and jogging all give body points.

Cleaning is gained by raking leaves.

I know the bowling alley description has Body listed... don't know that I've seen it in action. The electro-dance-sphere also builds Body.

Charisma can be learned also for both, older sim and toddler, if they repeatedly sing nursery rhyme together.
Ballet barre for body.
Random skill point sometimes after date or outing on community lot.

Even had a random skill on a home date. Also for private school.

Eggplant juice so long as the plant was healthy gives 1 random skill per glass.

Thanks ya'll. I've updated the list to include ya'lls replies.

Are there anymore we still missed?

Children playground items for body.

Oh, right, and the basketball hoop builds body, too. I don't know why anybody has sims work out - there's so many more fun ways to do it.

Children will usually gain one charisma point in the course of learning the nursery rhyme.

Doesn't raking the zen garden build something? Probably cleaning.

The uni coach can make sims workout.

Ok, new question add first post. Of all the items given, which ones will do they do autonomously (again unmodded or no cheats). I filled in what I can recall knowing or seein my sims doing. Any others I missed?

Curiously, the Vocophonic Sim Karaoke machine builds Creativity skill. (Shoudn't it be Charisma?) Leave one at an apartment lot, and watch the tenants max out their Creativity skill in no time.

I can't recall entirely since it's been a while since I played with the freestyle rap option you get by clicking on the active Sim from AL... or was it Uni? But I could have sworn that gave some Creativity as well? My game takes a long time to load so I can't double check as I was just about to sleep. Any chance someone else could verify this?

It also depends on expansion packs, of course, but:
They can gain skills by visiting community lots - outing or date. I've seen them gain cleaning, body, charisma and creativity points, depending on whom they hang out with.
They can gain body points with the bowling, yes - my military sims use that since they are too lazy to exercise.
Kids can gain body skills from some of the playground equipment (the play tower thing and I think one or two of the others too, though not the swings).
Knowledge Sims with the proper aspiration points can teach others any skill if they have high skills and the other Sim does not. I often use knowledge Sims to teach teenagers and children stuff.
Any Sim with high body points can train Sims with lower body points on most exercise equipment and the obstacle course, and it seems as if it is a bit faster than when they exercise on their own.
Autonomous - well, that depends on the objects on the lot - the piano and the telescope seem to have very high fun advertising levels to me.

The karaoke machine builds creativity in the game, charisma is a mod (by Neder, here on MTS).
Mechanical skills can be gained by the car wreck and I think also the model train table thingy (sorry, I cannot remember the names of the stuff).

Sims won't fix the car autonomously, but they can get a want to fix it.
Ballet bar has high advertising.
Chess
Exercise bike of doom, so autonomous it can kill.
Easel
all musical instruments
talking to a bird
toddler toys
Cooking channel slightly

Do kids learn cooking from the kid muffin stove?

Also, toddlers can gain charisma from autonomously singing the nursery rhyme alone. Just happened to a tot of mine recently.

Kids can yes. It's not a fast method.

I don't think you can earn mechanical from tinkering; I remember having to install a mod to do that very thing. Exercise machines are autonomous, like jo pointed out, the exercise bike can drain sims if you're not paying attention and sims love the ballet bar.

Oh, that reminds me, tai chi builds body and is autonomous; if your sim learns it, they'll do it often. And sims can (and will) join someone else doing tai chi or yoga autonomously, though I don't see many sims start yoga on their own too often.

You CAN get Mechanical points when tinkering - as long as you break the object being tinkered.

I don't understand why that exercise bike and ballet bar could kill. I placed several on my community lot, and the visitors ignore it. I don't use them on my home lot though - my sims are too poor to buy one anyway.

The bike, I think, was fixed or there is a mod that can fix it - it can kill in the sense that Sims may continue using them until they drop dead. But a player, of course, has to allow that to happen.

Sims can also gain charisma by working as a host at an owned restaurant, and body skill from waiting tables. The DJ booth builds creativity, too.

Autonomous:

Body - ballet bar, exercise bike, treadmill, bowling, jump rope, tai chi, playground items, football, basketball, electro dance sphere.
Charisma - bunny toy, nursery rhyme, talk to bird.
Cleaning - cleaning counters/toilets etc. Not sure about raking leaves and filling in holes - I've seen it happen, but it could be a mod thing.
Cooking - cooking meals, toy stove, Yummy channel.
Creativity - painting, musical instruments, karaoke, DJ booth, activity table, xylophone.
Logic - chess, telescope, crossword, toddler shape toy.
Mechanical - activity table.

@Ather, the Freestyle options come with Uni. I'm 85% sure. I'm 100% it's not AL or FT since Idon't have those EPs. Neither action raises charisma or creativity, only fun.

The exercise bike placed in an asylum challenge will kill sims because often they don't get off until needs are so low that they will go to bed instead of eat then they get out of bed and die of starvation. I've seen it on asylum challenges or people who don't know to check sims needs.